A Note on Congruences of Infinite Bounded Involution Lattices

Abstract

We prove that an infinite (bounded) involution lattice and even pseudo-Kleene algebra can have any number of congruences between 2 and its number of elements or equalling its number of subsets, regardless of whether it has as many ideals as elements or as many ideals as subsets. Furthermore, when they have at most as many congruences as elements, these involution lattices and even pseudo-Kleene algebras can be chosen such that all their lattice congruences preserve their involutions, so that they have as many congruences as their lattice reducts. Under the Generalized Continuum Hypothesis, this means that an infinite (bounded) involution lattice and even pseudo-Kleene algebra can have any number of congruences between 2 and its number of subsets, regardless of its number of ideals. Consequently, the same holds for antiortholattices, a class of paraorthomodular Brouwer-Zadeh lattices. Regarding the shapes of the congruence lattices of the lattice{ ordered algebras in question, it turns out that, as long as the number of congruences is not strictly larger than the number of elements, they can be isomorphic to any nonsingleton well-ordered set with a largest element of any of those cardinalities, provided its largest element is strictly join-irreducible in the case of bounded lattice-ordered algebras and, in the case of antiortholattices with at least 3 distinct elements, provided that the predecessor of the largest element of that well-ordered set is strictly join{irreducible, as well; of course, various constructions can be applied to these algebras to obtain congruence lattices with different structures without changing the cardinalities in question. We point out sufficient conditions for analogous results to hold in an arbitrary variety.